Narrow Results

The purpose of this work is to present an analytical description of dynamics of small ferromagnetic particles with uniaxial anisotropy and the slowly varying magnetic field applied at an arbitrary angle to the anisotropy axis. Considerable attention is given to the nonlinear aspects of adiabatic dynamics. Theoretical analysis based on the consideration of the Landau–Lifshitz–Gilbert equation employs an asymptotic expansion similar to the famous semi-classical WKBJ solution of quantum mechanics equations. The small parameter of the expansion is the ratio of characteristic frequency of the applied magnetic field to the precession frequency. The nonlinear equation describing the slow dynamics of small ferromagnetic particles is derived. The applicability conditions of the developed theory are presented. The nonlinear corrections to the precessional frequency are derived. It is shown that in numerical simulation using the Landau–Lifshitz–Gilbert equation it is not possible to neglect precessional terms at long simulation times due to the nonlinear dynamic correction to a position of the orbit center.

In this paper, ferromagnetic Co₅₀Fe₅₀ nanorings were fabricated by template-assisted sputtering deposition. The obtained Co₅₀Fe₅₀ nanorings have a good dimensional uniformity, with outer diameter of 360 nm, width of 50 nm and thickness of 40 nm. The magnetic properties of Co₅₀Fe₅₀ nanorings were investigated by micromagnetic simulations and vibrating sample magnetometer. We observed two parallel magnetic switching processes in the micromagnetic simulations of nanorings, the vortex formation process and onion rotation process, respectively. Quantitative analysis on the basis of experimental results indicates that 43.7% of the Co₅₀Fe₅₀ nanorings undergo the vortex formation process and others undergo the onion rotation process.

We study a thin-shell limit of micromagnetic energy for soft small ferromagnets. The relations between thickness of the magnet t, diameter l and magnetic exchange length w are t/l → 0 and tl/w² ≲ 1. We prove a Γ-convergence of the original 3D problem to a nonlocal 2D problem.

Existence of weak solutions is proved for a system of nonlinear parabolic equations/inequalities describing evolution of magnetization, temperature, magnetic field, and electric field in electrically-conductive unsaturated ferromagnets. The system is derived from a recently-proposed thermodynamically-consistent continuum theory for the ferro/paramagnetic transition. Besides the standard viscous-like damping, dissipation due to eddy currents and domain-wall pinning is considered.

We derive a Kronecker product approximation for the micromagnetic long-range interactions in a collocation framework by means of separable sinc quadrature. Evaluation of this operator for structured tensors (Canonical format, Tucker format, Tensor Trains) scales below linear in the volume size. Based on efficient usage of FFT for structured tensors, we are able to accelerate computations to quasi-linear complexity in the number of collocation points used in one dimension. Quadratic convergence of the underlying collocation scheme as well as exponential convergence in the separation rank of the approximations is proved. Numerical experiments on accuracy and complexity confirm the theoretical results.

Various applications ranging from spintronic devices, giant magnetoresistance sensors, and magnetic storage devices, include magnetic parts on very different length scales. Since the consideration of the Landau–Lifshitz–Gilbert equation (LLG) constrains the maximum element size to the exchange length within the media, it is numerically not attractive to simulate macroscopic parts with this approach. On the other hand, the magnetostatic Maxwell equations do not constrain the element size, but cannot describe the short-range exchange interaction accurately. A combination of both methods allows one to describe magnetic domains within the micromagnetic regime by use of LLG and also considers the macroscopic parts by a nonlinear material law using the Maxwell equations. In our work, we prove that under certain assumptions on the nonlinear material law, this multiscale version of LLG admits weak solutions. Our proof is constructive in the sense that we provide a linear-implicit numerical integrator for the multiscale model such that the numerically computable finite element solutions admit weak H¹-convergence (at least for a subsequence) towards a weak solution.

In this paper, we study the thin-film limit of the micromagnetic energy functional in the presence of bulk Dzyaloshinskii–Moriya interaction (DMI). Our analysis includes both a stationary $\mathrm{\Gamma }$-convergence result for the micromagnetic energy, as well as the identification of the asymptotic behavior of the associated Landau–Lifshitz–Gilbert equation. In particular, we prove that, in the limiting model, part of the DMI term behaves like the projection of the magnetic moment onto the normal to the film, contributing this way to an increase in the shape anisotropy arising from the magnetostatic self-energy. Finally, we discuss a convergent finite element approach for the approximation of the time-dependent case and use it to numerically compare the original three-dimensional (3D) model with the 2D thin-film limit.

Ferromagnetic materials are governed by a variational principle which is nonlocal, nonconvex and multiscale. The main object is given by a unit-length three-dimensional vector field, the magnetization, that corresponds to the stable states of the micromagnetic energy. Our aim is to analyze a thin film regime that captures the asymptotic behavior of boundary vortices generated by the magnetization and their interaction energy. This study is based on the notion of “global Jacobian” detecting the topological defects that a priori could be located in the interior and at the boundary of the film. A major difficulty consists in estimating the nonlocal part of the micromagnetic energy in order to isolate the exact terms corresponding to the topological defects. We prove the concentration of the energy around boundary vortices via a $\mathrm{\Gamma }$-convergence expansion at the second order. The second-order term is the renormalized energy that represents the interaction between the boundary vortices and governs their optimal position. We compute the expression of the renormalized energy for which we prove the existence of minimizers having two boundary vortices of multiplicity $1$. Compactness results are also shown for the magnetization and the corresponding global Jacobian.

The thermodynamic model of viscoelastic deformable magnetic materials at finite strains is formulated in a fully Eulerian way in rates. The Landau theory applied to a ferro-to-paramagnetic phase transition, the gradient theory (due to an exchange energy) for magnetization with general mechanically dependent coefficient, hysteresis in magnetization evolution by the Gilbert equation involving an objective corotational time derivative of magnetization, and the demagnetizing field are considered in the model. The Kelvin–Voigt viscoelastic rheology with a higher-order viscosity (exploiting the concept of multipolar materials) is used, allowing for physically relevant frame-indifferent stored energies and for local invertibility of deformation. The model complies with energy conservation and Clausius–Duhem entropy inequality. An existence and a certain regularity of weak solutions are proved by a Faedo–Galerkin semi-discretization and a suitable regularization.

We derive four reduced two-dimensional models that describe, at different spatial scales, the micromagnetics of ultrathin ferromagnetic materials of finite spatial extent featuring perpendicular magnetic anisotropy and interfacial Dzyaloshinskii–Moriya interaction. Starting with a microscopic model that regularizes the stray field near the material’s lateral edges, we carry out an asymptotic analysis of the energy by means of $\mathrm{\Gamma }$-convergence. Depending on the scaling assumptions on the size of the material domain versus the strength of dipolar interaction, we obtain a hierarchy of the limit energies that exhibit progressively stronger stray field effects of the material edges. These limit energies feature, respectively, a renormalization of the out-of-plane anisotropy, an additional local boundary penalty term forcing out-of-plane alignment of the magnetization at the edge, a pinned magnetization at the edge, and, finally, a pinned magnetization and an additional field-like term that blows up at the edge, as the sample’s lateral size is increased. The pinning of the magnetization at the edge restores the topological protection and enables the existence of magnetic skyrmions in bounded samples.

In this survey, we present several results on the regularizing effect, rigidity and approximation of 2D unit-length divergence-free vector fields. We develop the concept of entropy (coming from scalar conservation laws) in order to analyze singularities of such vector fields. In particular, based on entropies, we characterize lower semicontinuous line-energies in 2D and we study by Γ-convergence method the associated regularizing models (like the 2D Aviles–Giga and the 3D Bloch wall models). We also present some applications to the analysis of pattern formation in micromagnetics. In particular, we describe domain walls in the thin ferromagnetic films (e.g. symmetric Néel walls, asymmetric Néel walls, asymmetric Bloch walls) together with interior and boundary vortices.

The use of polarized currents as tools for influencing the magnetic switching of magnetic materials was an important step in the evolution of modern recording technologies. A new paradigm was opened and new technologies based on this new idea have been developed. However, the modeling effort did not followed at the same pace the technological developments. In this paper, we are analyzing the effect on switching critical curves of the polarized current, which seems to be the most sensitive tool for the moment in evidencing these effects. The most important point of this paper is that we took into consideration the effect of higher order terms in the series expansion of the uniaxial anisotropy free energy and we analyze how these terms are influencing the critical curve shape. The results are discussed in terms of the efficiency of the polarized currents on the switching behavior of macrospins.

The Spin-Cherenkov effect (SCE) refers to the Cherenkov-type excitation of spin waves (SWs). The Dzyaloshinskii–Moriya interaction (DMI) can cause chiral-symmetry breaking of the magnetization dynamics in magnetic systems, resulting in asymmetric dispersion relations and relevant effects. Here, we report a correlated phenomenon for Cherenkov SW excitations in a transversely magnetized nanostrip. In the presence of DMI, when a moving source exceeds the propagation velocity of SWs, coherent SWs are excited with different velocities in opposite directions. The minimum phase velocity of SWs is significantly reduced in one direction and increased in another. It is shown that the Cherenkov emission of SWs exhibits directional dependence, manifesting nonreciprocity which is related to the values of DMI constant D. Micromagnetic simulations along with analytical calculations are presented.

Arrays of interconnected magnetic nanorings show emergent dynamics arising from stochastic interactions at junctions between rings. These global emergent properties have already shown potential as a platform for reservoir computing. However, their experimental computational performance has been limited by the use of a single-dimensional global measurement, their magnetoresistance, as an output, masking much of the underlying state information. Here, we show in both simulation and experiment the potential of using the spin wave spectra of such arrays of nanorings to produce a fingerprint of the number and orientation of the internal magnetic states, providing a route towards a multi-dimensional output with which to perform computation. We show that the spectra fulfill the two criteria for reservoir computing: a nonlinear response to an input stimulus, and a fading memory.